ON THE 2ND ADJUNCTION MAPPING - THE CASE OF A 1-DIMENSIONAL IMAGE

Let (L) over cap be a very ample line bundle on an n-dimensional proje ctive manifold (X) over cap, i.e., assume that (L) over cap approximat e to iO-PN(1) for some embedding i : (X) over cap hooked right arrow P-N. In this article, a study is made of the meromorphic map, <(phi)ov er cap> : (X) over cap --> Sigma, associated to \K-(X over cap) + (n - 2)(L) over cap\ in the case when the Kodaira dimension of K-(X over c ap) + (n - 2)(L) over cap is greater than or equal to 3, and <(phi)ove r cap> has a 1-dimensional image. Assume for simplicity that n = 3. Th e first main result of the paper shows that <(phi)over cap> is a morph ism if either h(0)(K-(X over cap) + (L) over cap) greater than or equa l to 7 or kappa((X) over cap) greater than or equal to 0. The second m ain result of this paper shows that ii kappa((X) over cap) greater tha n or equal to 0, then the genus, g(f), of a fiber, f, of the map induc ed by <(phi)over cap> on hyperplane sections is less than or equal to 6. Moreover, if h(0)(K-(X over cap) + (L) over cap) greater than or eq ual to 21 then g(f) less than or equal to 5, a connected component F o f a general fiber of <(phi)over cap> is either a K3 surface or the blo wing up at one point of a K3 surface, and h(1)(O-(X over cap))) less t han or equal to 1. Finally the structure of the finite to one part of the Remmert-Stein factorization of <(phi)over cap> is worked out.